How do physicists think about Zeno's arrow?

[marplots]

So this is the second recommendation I can give the OP, aka Marsplot.

I thank you both for both.

 

[asydhouse]

It sounds like both arrows would suffer from the same defect, so that trying the experiment, at some level of fidelity, you wouldn't be able to detect a difference between the two. Or, is that wrong, and they wouldn't be equally "fuzzy?"

In other words, the fact that they can't be measured well seems to imply the differences between the two fade away - which, while it might help ruin the question, makes it even more mysterious that one still retains the "moving" identity and the other does not.
I'm pretty sure I've made an error in those two paragraphs, but I'm not sure what the error actually is.

I know you have mentioned "frames of reference" already, so my very amateur remark may be "duh" quality, but the supposedly unmoving arrow is of course moving very rapidly along with the planet, so this poetically mystical quality of "movingness" you are ascribing to the "moving" arrow is really a non-substance, since both arrows in fact are enmeshed in the quality of "movingness".

In other words, Tubbablubba has already cut through our layman's fog with mathematical insight, and this chasing after the Carrollian Snark of a mystical substance really is just a figment of your overattentive yearning for "something" which isn't there!

 

[W.D.Clinger]

It's a fair point that infinitesimals can be well-defined nowadays,

The standard reference for this is

Abraham Robinson. Non-standard Analysis. Revised Edition. Princeton University Press, 1996. (The original edition was published in 1974.)​

From the Foreward (1996), written by Wilhelmus A J Luxemburg:

....In the early history of the calculus, arguments involving infinitesimals played a fundamental role in the derivation of the basic rules of Newton's method of fluxions. The notion of an infinitesimal, however, lacked a precise mathematical definition, and soon their widespread use came under severe attack....infinitesimal reasoning gradually lost ground and survived only as a figure of speech. In fact, infinitesimals were gradually replaced by the D'Alembert-Cauchy concept of a limit to provide a firm foundation for the calculus. The creation of a mathematical theory of infinitesimals, as envisaged by Leibniz, on which to base the calculus remained an open problem.

Despite the discovery of non-archimedean order field extensions of the reals around the turn of the century, Leibniz' problem remained unsolved and lay dormant until the end of the 1950s....

In 1958, the aerodynamicist C. Schmieden, in collaboration with Laugwitz, constructed a partially order ring extension of the reals that included elements that could be viewed to play the role of infinitesimals....

At the end of the 1950s, however, unaware of the Schieden-Laugwitz approach, Abraham Robinson showed that the ordered fields that are non-standard models of the theory of real numbers could be viewed in the meta-mathematical or external sense as non-archimedean ordered field extensions of the reals that externally contain numbers that behave like infinitesimals....

From Robinson's preface to the first edition of the book:

In the fall of 1960 it occurred to me that the concepts and methods of contemporary Mathematical Logic are capable of providing a suitable framework for the development of the Differential and Integral Calculus by means of infinitely small and infinitely large numbers.....The resulting subject was called by me Non-standard Analysis since it involves and was, in part, inspired by the so-called Non-standard models of Arithmetic whose existence was first pointed out by T. Skolem.

From Robinson's preface to the Revised Edition of the book (1996):

Although the situation may change some day, the non-standard methods that have been proposed to date are conservative relative to the commonly accepted principles of mathematics...This signifies that a non-standard proof can always be replaced by a standard one, even though the latter may be more complicated and less intuitive....

A more definite opinion has been expressed in a statement which was made by Kurt Gödel after a talk that I gave in March 1973 at the Institute for Advanced Study, Princeton. The statement is reproduced here with Professor Gödel's kind permission.

'I would like to point out a fact that was not explicitly mentioned by Professor Robinson, but seems quite important to me; namely that non-standard analysis frequently simplifies substantially the proofs, not only of elementary theorems, but also of deep results....This state of affairs should prevent a rather common misinterpretation of non-standard analysis, namely the idea that it is some kind of extravagance or fad of mathematical logicians. Nothing could be farther from the truth. Rather there are good reasons to believe that non-standard analysis, in some version or other, will be the analysis of the future.

....Arithmetic starts with the integers and proceeds by successively enlarging the number system by rational and negative numbers, irrational numbers, etc. But the next quite natural step after the reals, namely the introduction of infinitesimals, has simply been omitted. I think, in coming centuries it will be considered a great oddity that the first exact theory of infinitesimals was developed 300 years after the invention of the differential calculus....'

The ellipses in quotations above show where I have shortened the quotations for this audience.

 

[Slings and Arrows]

Arithmetic starts with the integers and proceeds by successively enlarging the number system by rational and negative numbers, irrational numbers, etc. But the next quite natural step after the reals, namely the introduction of infinitesimals, has simply been omitted. I think, in coming centuries it will be considered a great oddity that the first exact theory of infinitesimals was developed 300 years after the invention of the differential calculus.

That’s the part of the story that I find so astonishing. Leibniz based the calculus on infinitesimals, which soon after fell into disrepute. Then nearly three centuries later, Robinson brings the infinitesimal back from the grave. And all the while, the calculus is in wide use and unfazed by all this behind the scene philosophical wrangling going on. It’s a fascinating story.

 

[TheAdversary]

'It's a fascinating story.'

Indeed. Maybe rigour is overrated. Calculus just worked.
But wasn't Robinson able to show that Newton and Leibniz were simply handling infinitesimals correctly?
It seems a rigorous framework isn't a requirement to get things done, but it can be a very useful tool when it's there.

 

[Darwin123]

'It's a fascinating story.'

Indeed. Maybe rigour is overrated. Calculus just worked.
But wasn't Robinson able to show that Newton and Leibniz were simply handling infinitesimals correctly?
It seems a rigorous framework isn't a requirement to get things done, but it can be a very useful tool when it's there.

Yes.

This answers the OP, at least partly. Many physicists think of Zeno's paradoxes as the very beginning of calculus. The mathematical tools of calculus are often seen as 'resolving' Zeno's paradox to an extent.


There was more than one Zeno's paradox, also. They all dealt with a concept very much like calculus, in one way or another.

I learned one thing from this thread. I didn't realize that infinitesimal and topology are two lines of reasoning that converged. They are logically equivalent, but they have superficial differences. One could think of both concepts as representations of the same theory that together is now called calculus.

 

[TubbaBlubba]

Something that Maxwell, I believe, once said struck me - physical laws describe not quantities, but relationships between quantities. Essentially, in order to understand physics, you must accept certain quantities as primitive notions, their nature understood by intuition. In Newtonian mechanics for instance, this applies to "position", "mass" and "time", from wich other quantities can be derived using calculus. In other fields you have more lofty quantities like "energy", "charge" or even "field density", which may require quite a few leaps of imagination to grasp.

 

[Darwin123]

I thank you both for both.

Basically, there are two ways to derive calculus. Both of them shed some light on Zeno’s Paradoxes.

https://en.wikipedia.org/wiki/Non-standard_analysis
‘The history of calculus is fraught with philosophical debates about the meaning and logical validity of fluxions or infinitesimal numbers. The standard way to resolve these debates is to define the operations of calculus using epsilon–delta procedures rather than infinitesimals. Non-standard analysis[1][2][3] instead reformulates the calculus using a logically rigorous notion of infinitesimal numbers.
Non-standard analysis was originated in the early 1960s by the mathematician Abraham Robinson.’



http://users.cms.caltech.edu/~jtropp/papers/Tro99-Infinitesimals.pdf
‘Infinitesimals have enjoyed an extensive and scandalous history. Al- most as soon as the Pythagoreans suggested the concept 2500 years ago, Zeno proceeded to drown it in paradox. Nevertheless, many mathema- ticians continued to use infinitesimals until the end of the 19th century because of their intuitive appeal in understanding continuity. When the foundations of calculus were formalized by Weierstrass, et al. around 1872, they were banished from mathematics.
As the 20th century began, the mathematical community officially regarded infinitesimals as numerical chimeras, but engineers and physi- cists continued to use them as heuristic aids in their calculations. In 1960, the logician Abraham Robinson discovered a way to develop a rigorous theory of infinitesimals.’